Found problems: 966
2012 Singapore MO Open, 3
For each $i=1,2,..N$, let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$.
1958 November Putnam, B5
The lengths of successive segments of a broken line are represented by the successive terms of the harmonic progression $1, 1\slash 2, 1\slash 3, \ldots.$ Each segment makes with the preceding a given angle $\theta.$ What is the distance and what is the direction of the limiting points (if there is one) from the initial point of the first segment?
1980 Putnam, A6
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$
and $f(1)=1 .$ Determine the largest real number $u$ such that
$$u \leq \int_{0}^{1} |f'(x) -f(x) | \, dx $$
for all $f$ in $C.$
2006 Putnam, B2
Prove that, for every set $X=\{x_{1},x_{2},\dots,x_{n}\}$ of $n$ real numbers, there exists a non-empty subset $S$ of $X$ and an integer $m$ such that
\[\left|m+\sum_{s\in S}s\right|\le\frac1{n+1}\]
2003 Putnam, 1
Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?
1975 Putnam, B5
Define $f_{0}(x)=e^x$ and $f_{n+1}(x)=x f_{n}'(x)$. Show that $\sum_{n=0}^{\infty} \frac{f_{n}(1)}{n!}=e^e$.
2013 Putnam, 4
For any continuous real-valued function $f$ defined on the interval $[0,1],$ let \[\mu(f)=\int_0^1f(x)\,dx,\text{Var}(f)=\int_0^1(f(x)-\mu(f))^2\,dx, M(f)=\max_{0\le x\le 1}|f(x)|.\] Show that if $f$ and $g$ are continuous real-valued functions defined on the interval $[0,1],$ then \[\text{Var}(fg)\le 2\text{Var}(f)M(g)^2+2\text{Var}(g)M(f)^2.\]
1963 Putnam, B6
Let $E$ be a Euclidean space of at most three dimensions. If $A$ is a nonempty subset of $E$, define $S(A)$ to be the set of points that lie on closed segments joining pairs of points of $A$ (a one-point set should be considered to be a special case of a closed segment). For a given nonempty set $A_0$, define $A_n =S(A_{n-1})$ for $n=1,2,\ldots$ Prove that $A_2 =A_3 =\ldots.$
1970 Putnam, B3
A closed subset $S$ of $\mathbb{R}^{2}$ lies in $a<x<b$. Show that its projection on the $y$-axis is closed.
2019 Putnam, A6
Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on the open interval $(0,1)$. Suppose that for some real number $r>1$,
\[
\lim_{x\to 0^+}\frac{g(x)}{x^r} = 0.
\]
Prove that either
\[
\lim_{x\to 0^+}g'(x) = 0\qquad\text{or}\qquad \limsup_{x\to 0^+}x^r|g''(x)|= \infty.
\]
1965 Putnam, B6
If $A$, $B$, $C$, $D$ are four distinct points such that every circle through $A$ and $B$ intersects (or coincides with) every circle through $C$ and $D$, prove that the four points are either collinear (all on one line) or concyclic (all on one circle).
Putnam 1939, B3
Given $a_n = (n^2 + 1) 3^n,$ find a recurrence relation $a_n + p a_{n+1} + q a_{n+2} + r a_{n+3} = 0.$ Hence evaluate $\sum_{n\geq0} a_n x^n.$
1956 Putnam, A5
Call a subset of $\{1,2,\ldots, n\}$ [i]unfriendly[/i] if no two of its elements are consecutive. Show that the number of unfriendly subsets with $k$ elements is $\binom{n-k+1}{k}.$
2004 Putnam, B5
Evaluate $\lim_{x\to 1^-}\prod_{n=0}^{\infty}\left(\frac{1+x^{n+1}}{1+x^n}\right)^{x^n}$.
1941 Putnam, A3
A circle of radius $a$ rolls in the plane along the $x$-axis. Show that the envelope of a diameter is a cycloid, coinciding with the cycloid traced out by a point on the circumference of a circle of diameter $a$, likewise rolling in the plane along the $x$-axis.
2016 District Olympiad, 4
Let $ f:[0,1]\longrightarrow [0,1] $ be a nondecreasing function. Prove that the sequence
$$ \left( \int_0^1 \frac{1+f^n(x)}{1+f^{1+n} (x)} \right)_{n\ge 1} $$
is convergent and calculate its limit.
1973 Putnam, A3
Let $n$ be a fixed positive integer and let $b(n)$ be the minimum value of
$$k+\frac{n}{k},$$
where $k$ is allowed to range through all positive integers. Prove that $\lfloor b(n) \rfloor= \lfloor \sqrt{4n+1} \rfloor.$
1940 Putnam, A8
A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$.
Show that its area, disregarding sign, is
$$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$
where $\Delta$ is the discriminant of the matrix
$$M=\begin{pmatrix}
a_1 & b_1 &c_1\\
a_2 & b_2 &c_2\\
a_3 & b_3 &c_3
\end{pmatrix}.$$
1995 Putnam, 1
For a partition $\pi$ of $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$, let $\pi(x)$ be the number of elements in the part containing $x$. Prove that for any two partitions $\pi$ and $\pi^{\prime}$, there are two distinct numbers $x$ and $y$ in $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ such that $\pi(x) = \pi(y)$ and $\pi^{\prime}(x) = \pi^{\prime}(y)$.
1959 Putnam, A2
"Let $\omega^3 = 1, \omega \neq 1$. Show that$z_1, z_2, -\omega z_1, -\omega^2z_2$ are the vertices of an equilateral triangle."
1980 Putnam, B2
Let $S$ be the solid in three-dimensional space consisting of all points $(x,y,z)$ satisfying the following six
simultaneous conditions:
$$ x,y,z \geq 0, \;\; x+y+z\leq 11, \;\; 2x+4y+3z \leq 36, \;\; 2x+3z \leq 44.$$
a) Determine the number $V$ of vertices of $S.$
b) Determine the number $E$ of edges of $S.$
c) Sketch in the $bc$-plane the set of points $(b, c)$ such that $(2,5,4)$ is one of the points $(x, y, z)$ at which the linear function $bx + cy + z$ assumes its maximum value on $S.$
1997 Flanders Math Olympiad, 1
Write the number 1997 as the sum of positive integers for which the product is maximal, and prove there's no better solution.
1987 Putnam, A3
For all real $x$, the real-valued function $y=f(x)$ satisfies
\[
y''-2y'+y=2e^x.
\]
(a) If $f(x)>0$ for all real $x$, must $f'(x) > 0$ for all real $x$? Explain.
(b) If $f'(x)>0$ for all real $x$, must $f(x) > 0$ for all real $x$? Explain.
2017 Putnam, B1
Let $L_1$ and $L_2$ be distinct lines in the plane. Prove that $L_1$ and $L_2$ intersect if and only if, for every real number $\lambda\ne 0$ and every point $P$ not on $L_1$ or $L_2,$ there exist points $A_1$ on $L_1$ and $A_2$ on $L_2$ such that $\overrightarrow{PA_2}=\lambda\overrightarrow{PA_1}.$
2020 Putnam, A4
Consider a horizontal strip of $N+2$ squares in which the first and the last square are black and the remaining $N$ squares are all white. Choose a white square uniformly at random, choose one of its two neighbors with equal probability, and color tis neighboring square black if it is not already black. Repeat this process until all the remaining white squares have only black neighbors. Let $w(N)$ be the expected number of white squares remaining. Find
\[ \lim_{N\to\infty}\frac{w(N)}{N}.\]